Subgroups — #15691652

Order 3159.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11760454	C₃	1	3	yes	maximal, complemented
2	C₃×C₉×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#11760443	C₃	1	1	yes	maximal, characteristic, complemented
4	#11760441	C₃	1	1	yes	maximal, characteristic, complemented
5	#11760439	C₃	1	3	yes	maximal, complemented
6	#11760436	C₃	1	3	yes	maximal, complemented
7	C₁₃×#733	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
9	C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃×C₉×C₁₃	C₃²	1	1	yes	characteristic
11	C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
12	#66851	C₃²	1	3	yes	—
13	C₃×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
15	#66849	—	3	3	no	—
16	#66848	—	3	3	no	—
17	C₉×C₁₃⋊C₃	—	3	1	no	—
18	C₉×C₁₃⋊C₃	—	3	1	no	—
19	C₉×C₁₃⋊C₃	—	3	1	no	—
20	C₉×C₁₃⋊C₃	—	3	3	no	—
21	#6908	—	13	1	no	maximal, Sylow
22	C₃²×C₁₃	C₃³	1	1	yes	characteristic
23	C₃×C₁₃⋊C₃	—	3	1	no	—
24	C₃×C₁₃⋊C₃	—	3	3	no	—
25	C₃×C₁₃⋊C₃	—	3	3	no	—
26	C₉×C₁₃	—	3	1	no	—
27	C₉×C₁₃	—	3	3	no	—
28	C₁₃⋊C₉	—	3	1	no	—
29	C₁₃⋊C₉	—	3	3	no	—
30	C₁₃⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
31	C₁₃⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
32	C₁₃⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
33	#737	—	13	3	no	—
34	C₃²×C₉	—	13	1	no	—
35	#733	—	13	1	no	—
36	#733	C₁₃⋊C₃	1	1	yes	characteristic, complemented
37	#732	—	13	1	no	—
38	#731	—	13	3	no	—
39	AΣL(1,𝔽₂₇)	—	13	3	no	—
40	C₃×C₁₃	—	3	1	no	—
41	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₁₃⋊C₃	—	3	3	no	—
43	C₁₃⋊C₃	—	9	3	no	—
44	C₃³	—	13	1	no	—
45	3₋¹⁺²	—	13	3	no	—
46	3₋¹⁺²	—	39	3	no	—
47	3₋¹⁺²	—	39	3	no	—
48	3₋¹⁺²	C₃×C₁₃⋊C₃	1	3	yes	—
49	H(3,𝔽₃)	—	13	3	no	—
50	C₃×C₉	—	13	1	no	—
51	C₃×C₉	—	13	1	no	—
52	C₃×C₉	—	39	1	no	—
53	C₃×C₉	—	39	1	no	—
54	C₃×C₉	—	39	1	no	—
55	C₃×C₉	—	39	3	no	—
56	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
57	C₁₃	#6908	1	1	yes	characteristic, complemented, Sylow
58	C₃²	—	39	3	no	—
59	C₃²	—	39	1	no	—
60	C₃²	—	39	3	no	—
61	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
62	C₉	—	3	1	no	—
63	C₉	—	3	3	no	—
64	C₉	—	13	1	no	—
65	C₉	—	13	1	no	—
66	C₉	—	13	1	no	—
67	C₉	—	39	1	no	—
68	C₉	—	39	3	no	—
69	C₃	—	3	1	no	—
70	C₃	—	39	3	no	—
71	C₃	—	117	3	no	—
72	C₃	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₃⋊C₃
commutator, cosocle	C₃²×C₁₃	C₃³
Frattini	C₃²	C₃²×C₁₃⋊C₃
Fitting	C₁₃×#733	C₃
socle	C₃×C₁₃	C₃×H(3,𝔽₃)
hypercenter	#733	C₁₃⋊C₃
hypocenter	C₁₃	#6908
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #11760454	C₃	semidirect
2. C₃×C₉×C₁₃⋊C₃	C₃	semidirect
3. #11760443	C₃	semidirect
4. #11760441	C₃	semidirect
5. #11760439	C₃	semidirect
6. #11760436	C₃	semidirect
7. C₁₃×#733	C₃	semidirect
36. #733	C₁₃⋊C₃	semidirect
57. C₁₃	#6908	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11760454	3
C₃×C₉×C₁₃⋊C₃	3
#11760443	3
#11760441	3
#11760439	3
#11760436	3
C₁₃×#733	3
#6908	13